Do abilities that trigger upon death work if the character is blanked? Do the cards enter the discard pile at the same time or does one go in before another? Or is it a active or first player decision to determine which card hits a discard pile first?

When multiple cards are discarded for an effect, the discard occurs simultaneously, so they would enter the discard pile simultaneously.

Attachments 'leave play' and are discarded simultaneous to the attached character leaving play. Let's look at Cloud the Mind and Sincerity - cloud the mind is no longer attached 'when' the character leaves play. Sincerity triggers 'when' the card leaves play. Because both happen at the transition from in-play to out-of-play, I would say that you do not get to resolve the keyword.

I think there was a recent ruling for another card when there was a 'transition', I think that may help, but I can't remember what it was.

At the same time. The card is still "blank" when it is discarded.

Current Rules Reference, page 2, 4th bullet point under Attachments:

Quote

If the card to which an attachment is attached leaves play, simultaneously discard the attachment.